Abstract

We introduce a class of finite-dimensional algebras built from a partial order generated as a transitive relation from a finite tree. These algebras, known as tree algebras, have the property that every locally contractive representation has a *-dilation. Furthermore, they satisfy an appropriate analogue of the Sz. Nagy–Foia Commutant Lifting Theorem. Then we define the infinite-dimensional analogue of these algebras in the class of completely distributive CSL algebras. These algebras are shown to have the semidiscreteness and complete compact approximation properties with respect to the class of finite-dimensional tree algebras. Consequently, they also have the property that contractive weak-* continuous representations have *-dilations, and satisfy the Sz. Nagy–Foia Commutant Lifting Theorem.